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On class groups of imaginary quadratic fields
Author(s) -
Wiles A.
Publication year - 2015
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdv031
Subject(s) - mathematics , the imaginary , quadratic equation , class (philosophy) , prime (order theory) , trace (psycholinguistics) , quadratic field , galois module , pure mathematics , elliptic curve , modular form , class field theory , algebra over a field , modular design , arithmetic , discrete mathematics , algebraic number field , combinatorics , quadratic function , geometry , computer science , psychology , linguistics , philosophy , artificial intelligence , psychotherapist , operating system
In this paper, it is proved that one can find imaginary quadratic fields with class number not divisible by a specified prime l and with certain specified splitting conditions at a finite number of primes. Such existence theorems are useful in the arithmetic of elliptic curves and, potentially, also in certain lifting problems for reducible two‐dimensional Galois representations. The methods used are a blend of geometry and the theory of modular forms, especially the trace formula.